J . Phys. Chem. 1990, 94, 3454-3461
3454
TABLE VII: Components of the Interaction Energy for the Linear NO-('Z-).-Ar('S) Complex in the van der Waals Minimum Region R-..,. = 3.25 k A r basis [7s4p2d] 6,, 6No[7s4p2dlf] 6,, aNOAESCF -1231.2 -329.0 -57.7 -1268.9 -338.6 -93.0 AE(2' -906.5 -402.5 -499.4 -1047.2 -541.0 -561.5 AE(2) -2137.7 -731.3 -557.1 -2316.0 -879.5 -654.7 +66.8 -9.6 +40.3 SEN +40.5 + I 1.2 +8.1 AESDQ -3.0 +14.8 -13.0 AE(4) -135.6 -4.3 -83.8 AE(4) -2206.5 -745.1 -600.7
"The [532] basis was used for NO-. All energies corrected for BSSE. Energies in k E h . actually typical for complexes with Ar (cf. Ar2.13 MgAr,30 and NH3Ar3'). The UMP2 interaction energy of 2316 WE,,obtained with the [7s4p2dIf] basis for Ar may be considered as a lower bound of De in the NO-(31;-)-.Ar('S) complex. This value agrees fairly well with the experimental estimate of De,which is 0.058 eV (2100 p E h ) , obtained by Arnold et al.' We, however, expect that the exact value should be larger rather than lower with respect to our estimate. Conclusions
I t has been shown that the NO- anion may form weak van der Waals complexes with rare-gas atoms (Rg) He and Ar; the global (31)Chalasinski, G.; Szczesniak, M. M.; Cybulski, S. M.; Scheiner, S. J . Chem. Phys. 1989, 91. 7809.
minimum was found at the collinear configuration N-O--.Rg, for the 0-Rg distance about 3.25 8, and of the depth of 69 and 509 cm-' for He and Ar, respectively. Approximate scan of the potential energy surface for the NO-(32-).-He( IS)system may be helful to establish the geometry of other NO--.Rg complexes as well as NO- complexes with water and N 2 0 . It should also be stressed that the MPPT approach adopted to NO- and NO---Rg proved to be as satisfactory as for complexes of closed-shell atoms and molecules. It should be stressed that reliable evaluation of interaction energies was possible only by using the counterpoise method to correct for BSSE. Adequacy and efficiency of this technique have been recently pointed out in several review article^,'^.^^.^^ and its usefulness is continuously being c ~ n f i r m e d . ' ~ . ~ ~ , ~ ~ - ~ ~
Acknowledgment. G.C. thanks M. M. Szczesniak and S. Scheiner for their kind hospitality during his stay at Carbondale. We acknowledge the financial support of the National Institutes of Health (GM 36912) and Polish Academy of Sciences (the program CPBPOl.12.). Registry No. NO, 10102-43-9; Ar, 7440-37-1;He, 7440-59-7. (32)Chalasinski, G.; Gutowski, M. Chem. Reu. 1988, 88, 943. (33)Hobza, P.; Zahradnik, R. Chem. Reo. 1988,88, 871. (34)Diercksen, G. H. F.; Sadlej, A. Y. Chem. Phys. 1989, 131, 215. (35)Liu, B.; McLean, A. D. J . Chem. Phys. 1989, 91, 2348. (36)Alagona, G.; Ghio, C.; Cammi, R.; Tomasi, J. In Molecules in Physics, Chemistry and Biology; Maruani, J., Ed.; Kluwer Academic Publishers: Boston, 1988;Vol. 2, p 507. Alagona, G.; Ghio. C.; Tomasi, J. J . Phys. Chem. 1989, 93, 5401. (37)Chalasinski, G.; Simons, J. Chem. Phys. Lett. 1988, 148, 289. (38)van Lenthe, J. H.; Vos, R. J.; van Duijneveldt-van de Rijdt. J. G. C. M. Chem. Phys. Letf. 1988, 143, 435.
Triplet Excitation Transport Kinetics in Vapor-Deposited Naphthalene Laurel A. Harmon and Raoul Kopelman* Chemistry Department, The University of Michigan. Ann Arbor, Michigan 48109- 1055 (Received: June 6, 1989)
The phosphorescence and delayed fluorescence of vapor-deposited naphthalene films have been studied as functions of time. The decays are analyzed in terms of two distinct domain types: slightly perturbed crystalline domains and highly perturbed domains (domain boundaries). The latter regions exhibit a high degree of energetic disorder and a low-dimensional effective topology. The annihilation kinetics is shown to be "fractal-like", with a time-dependent reaction rate coefficient whose power ( h ) is found to be on the order of 0.5 (rather than the classical value of 0). Kinetic formulations are given for homofusion, heterofusion, and linear combinations thereof. Analytic expressions are derived for the time dependence of both phosphorescence and delayed fluorescence intensities.
1. Introduction
The kinetics of excitation transport in disordered or heterogeneous condensed phases'-I2 is playing an increasingly important role in a number of fields, such as heterogeneous catalysis and in vivo biological reactions.' Naphthalene, like other organic materials,* forms nonequilibrium solids with both structural and ( I ) Kopelman. R. Science 1988, 241, 1620. (2)BBssler, H. Phys. Sratus Solid B 1981, 107. 9. (3)Kopelman, R.;Klymko, P. W.; Newhouse, J. S.; Anacker, L. W. Phys. Reo. B 1984, 29, 3747. (4)Klymko, P. W.;Kopelman, R. J . Phys. Chem. 1983.87, 4565. ( 5 ) Prasad, J.; Parus, S. J.; Kopelman, R.Phys. Reu. Lett. 1986.56, 1742. (6) Newhouse. E. 1.; Kopelman, R. Chem. Phys. Lett. 1988, 143, 106. (7)Parson, R. P.; Kopelman, R. Chem. Phys. Lett. 1982, 87, 528. (8) Lin, Y.; Hanson, D. M. J . Phys. Chem. 1987, 91, 2279. (9)Ahlgren, D. C.;Monberg, E. M.; Kopelman, R. Chem. Phys. Lett. 1979, 64, 122. ( I O ) Ahlgren, D. C. Ph.D. Dissertation, The University of Michigan, 1979. ( 1 I ) Gentry, S. T.; Kopelman, R . J . Chem. Phys. 1984, 81, 3014.
energetic disorder, when deposited from vapor at low temperatures. Vapor-deposited naphthalene therefore provides an excellent model system in which to study the effects of disorder on excitation transport. Spectroscopic studies have shown that naphthalene ~ a prepared by vapor deposition is h e t e r o g e n e o ~ s . ' ~ - ' Within single sample, quasi-crystalline regions exist with little energetic disorder, referred to as type I regions. Forming the boundaries of such regions are areas of much greater structural disorder, characterized by an energy bandwidth of about 100 cm-' and designated type I1 regions. We present here an investigation of triplet-triplet annihilation reaction kinetics in type I1 regions of vapor-deposited naphthalene. ~
~~~~
(12) Gentry, S. T.; Kopelman, R . J . Chem. Phys. 1984, 81, 3022. ( I 3) Harmon, L. A. Ph.D. Dissertation, The University of Michigan, 1985. (14)Harmon, L. A.; Kopelman, R. J . Lumin. 1984, 31/32, 660. ( I 5) Harmon, L. A,; Kopelman, R. In Unconuentionaf Phoroactiue Solids; Scher, H., Ed.; Plenum Press: New York, 1988;p 83. (16) Harmon, L. A . ; Kopelman, R . Manuscript in preparation.
0022-3654/90/2094-3454$02.50/00 1990 American Chemical Society
Transport Kinetics in Vapor-Deposited Naphthalene The triplet-triplet annihilation reaction is well-established as a probe of excitation transport in crystalline and noncrystalline solids.Id Reported here are experimental phosphorescence and delayed fluorescence decays at 6 K. Analyses of the decays yield the effective molecularity of the annihilation reaction and permit characterization of the annihilation rate coefficient, k ( t ) . Information regarding transport in the medium is obtained from the time dependence of k , which follows a power law like that observed in other heterogeneous system^.^-^ The reaction under consideration is the fusion (annihilation) of two naphthalene triplet excitons to form a higher excited singlet which very rapidly decays to the first excited singlet:
TI
+ TI
The Journal of Physical Chemistry, Vol. 94, No. 9, I990 3455 1
FLUOR.
t 5 4 0 NM IC(
-swP 420
m
H EXCIWER
+
S*
-+
SI
Due to the short naphthalene singlet lifetime (120 ns'), the SI state rapidly decays. This decay is characterized by the emission of a photon (prompt fluorescence). However, the time scale of deiuyedfluorescence is determined by the time required for two triplet excitations to meet via migration in the medium and may be as long as the natural triplet-state lifetime (2.6 sell). In crystalline naphthalene, rapid triplet transport leads to delayed fluorescence on the scale of nanoseconds.1D-12Delayed fluorescence at longer times is attributed to a slowing of triplet excitation transport by substitutional, energetic, or structural disorder.)-I2 In order to isolate reaction kinetics in type I1 regions, delayed fluorescence and phosphorescence decays were monitored in the millisecond regime. The kinetics of triplet-triplet annihilation is examined in this work by analysis of the time dependence of delayed fluorescence and phosphorescence intensities. Classically, the triplet-triplet reaction rate coefficient is independent of time, and the two decays (annihilation and natural) can be combined in a simple way to establish the effective molecularity of the annihilation reaction. However, in lower dimensional or heterogeneous media, classical kinetics no longer hold. The annihilation rate coefficient is time-dependent, leading to a more complex relationship between the delayed fluorescence and phosphorescence decays. The general rate equation for the decay of a triplet population subject to both annihilation and natural decay is derived in section 3. Approximate solutions to the rate law are derived in section 4.1. The approximations are applied to experimental decays in section 5.1, and their limitations are discussed. Exact solutions to the rate equation are derived in section 4.2 and lead to a more general method for analyzing the annhilation kinetics from delayed fluorescence decays. In section 5.2, the results of fitting experimental delayed fluorescence decays to the analytical expressions of section 4.2 are presented. The details of the actual fitting procedure are included as an Appendix. The implications of the results are discussed in section 6.
2. Experimental Methods The naphthalene used in this work was extensively purified by zone refining and fusion with potassium metal. Samples were prepared by evaporation onto quartz substrates at temperatures between 50 and 70 K. The details of sample purification, preparation, and annealing have been described e l ~ e w h e r e . l ) - The ~~ naphthalene SI state was excited by 310-nm radiation from a 1600-W xenon arc lamp passed through a 0.25-111 Jobin-Yvon monochromator with 2-mm slits; the lower energy TI state was populated by intersystem crossing. Emission spectra were collected through a Jobin-Yvon I-m double monochromator with an EM1 9816QB photomultiplier and digitized with a PAR Model 1109 photon counter. The fluorescence (UV) and phosphorescence (green) of naphthalene single crystals are widely separated in wavelength. However, vapor-deposited naphthalene shows three types of emission in the visible region:I3J6 broad singlet excimer emission centered at about 24 500 cm-I, naphthalene (21 200-19 500 cm-I), and a naphthalene radical ( I -HNR) at 18 555 cm-I and below. Unless qualified, in what follows "phosphorescence" refers to total emission intensity collected in the visible region. Because the intensities of delayed emission resolved with the double mono-
sisod.o
zssod. o
msod. o
2550d. o
zssod.o
CY-I
__1 RADICAL
4740 A
ti
zisod. o
issod.o
I d .o
Figure 1. Steady-state spectrum of a freshly prepared 100%naphthalene-h, sample at 10 K showing the regions in which fluorescence (FLUOR), excimer, phosphorescence (PHOS), and 1 -HNR (Radical) are observed. Also indicated are the spectral regions transmitted by the various filters discussed in section 2.
chromator were found to be too low, spectral regions were selected with glass filters. The delayed fluorescence was collected through two Corning 7-54 filters which transmit the UV and eliminate phosphorescence. Phosphorescence decays were collected through two Corning 0-5 1 filters to block the UV. Narrower regions of the visible spectrum were selected with a single Corning 0-51 in conjunction with another filter. Delayed excimer emission was isolated with a 420-nm (23 800-cm-]) band-pass filter (10-nm bandwidth). Naphthalene phosphorescence was recorded with either a 500-nm short wave pass (SWP) filter, which transmits visible light above 20000 cm-I, or a 4740-A (21 100-cm-l) band-pass filter with a 70-A bandwidth. Decays recorded with the SWP filter contain both phosphorescence and delayed excimer emission; the tail of the excimer emission may even extend into the region selected with the 4740-A filter, but it appears from steady-state spectra to be weak. The regions in which these filters transmit are shown in Figure 1. To obtain delayed fluorescence and phosphorescence decays, the excitation beam was shuttered. Time-resolved intensities were monitored by an EM1 9781R photomultiplier in a room-temperature housing placed behind a shutter at the cryostat window. The phototube output was averaged with a PAR Model 4202 signal averager. In order to protect the phototube from lamp emission and prompt fluorescence, the shutter in the emission beam was triggered to open 3 ms after the excitation shutter closed. This distorted the first 10 ms of data which were therefore discarded. Each decay consists of 1024 points at 1-ms intervals, typically averaged over 100 scans. Scans were repeated at intervals of either 10 or 20 s to allow the triplet excitations to equilibrate during illumination and to decay to a negligible concentration. Background intensities were recorded before and after every decay. All spectra and decays were recorded on an LSI-I 1/03 laboratory computer. Spectra were smoothed on the LSI-I 1 and plotted by a Calcomp plotter in conjunction with the Ahmdal470 computer at The University of Michigan. Both fitting and plotting of the experimental delayed fluorescence decays were carried out on the Ahmdal 470.
3. The Rate Equation The intensity of delayed fluorescence at any time t , Idf(t), is proportional to the number of annihilation events, ndf(t), Idf(t) Ot ndf(t) (2) in which the proportionality constant is determined by the radiative quantum efficiency of the first singlet." The number of annihilation events is in turn given by (3)
3456
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
where p ( t ) is the density of triplet excitons. The triplet-triplet annihilation reaction rate in disordered systems has been explained in terms of the number of distinct sites, S ( t ) , visited by a random walker in equivalent media.I7 For random walks on fractal structures, the time dependence of S is given by 4,18-21
S ( t ) 0: tf
O
r f I~
Harmon and Kopelman nihilation and the second describes the effects of natural decay. Here k l is the unimolecular decay rate, related to the natural lifetime by T = k I - l . The form of the annihilation rate has been taken from eq 5. The units of k l and k ( t ) are (time)-'. Substitution of k ( t ) from eq 6 into eq 8 yields the following general rate equation:
(4)
The exponentfhas been related to a characteristic dimension of by f = dJ2. the medium, d,, the spectral dimen~ion,l**'~ The triplet-triplet annihilation reaction rate takes the form
Approximate and exact solutions to (9) are derived in sections 4.1 and 4.2 and applied to experimental data in section 5.
in which m, the effective molecularity of the annihilation reaction, has been left unspecified to include both binary ( m = 2) and unary ( m = 1 ) processes. Although annihilation is strictly a binary reaction, pseudo-first-order kinetics is observed when one excitation belongs to a population whose density is approximately time independent." In such cases, k ( t ) incorporates p', the density of the time-independent population. This factor of p' is implied for the m = 1 cases in all expressions generalized to cover both cases. Expressions reserved for pseudounary reactions will take explicit account of p'. The terms homofusion and heterofusion are used to refer to reactions with effective molecularities of 2 and 1, respectively. The triplet-triplet annihilation reaction rate coefficient, k ( t ) , is in turn related to S ( t ) by4jZ0
= kannt-h
h
f -1
in which h = 0 or f = I expresses the limit of a time-independent rate coefficient. This form of k ( t ) has proved to be characteristic of reactions occurring in low-dimensional, heterogeneous, or disordered materials.' It has been observed in simulations of reactions in energetically disordered media.z1.22 Experimental studies of exciton annihilation in substitutionally disordered naphthalene crystals near percolation,) porous glass and artificial membra ne^,^ and polymeric g l a s s e ~have ~ ? ~ also demonstrated the power law dependence of the rate coefficient on time. By combining eq 2, 3, 5, and 6, we see that the time dependence of the delayed fluorescence intensity is related to the time dependence of the triplet population by
I d f ( t ) and p ( f ) may be experimentally measured as functions of time via delayed fluorescence and phosphorescence decays, respectively. If h = 0, the two decays can be combined to establish the effective molecularity of the reaction, Le., the value of m. However, the time dependence of the annihilation rate coefficient in the case of reactions in lower dimensional or heterogeneous media requires a more thorough analysis of the rate law. In general, triplet populations are subject to both annihilation and natural decay, with a time dependence given by
-2 = dt =
-(;),,
-
(g)
k(t)pm+ k , p
nd
(8)
in which the first term describes the depletion of triplets by an(17) Montroll. E. W.; Weiss, G . H.J . Marh. Phys. 1965, 6 , 1667. (18) Alexander, S.;Orbach, T. J . Phys. Lett. 1982, 43, L625. (19) Rammal, R.; Toulouse, G. J . Phys. Lett. 1982, 44, L13. (20) de Gennes, P. G. C.R . Acad. Sci., Ser. B 1983, 296, 881. (21) Anacker, L. W.; Kopelman, R.; Newhouse, J . S.J . Star. Phys. 1984,
36, 591. (22) Schonherr, G.; Eiermann, R.; Bassler. H.; Silver, M . Chem. Phys. 1980, 52. 287.
4. Solutions to the Rate Equation 4.1. Approximate Solutions. Equation 7 relates the time dependence of the delayed fluorescence intensity, I d f ( t ) , to that of the triplet population density, p ( t ) . This expression assumes that annihilation occurs within a single population. At times short relative to the natural lifetime, or if only a small fraction of triplets annihilates, p ( t ) may be approximated as a constant. Equation 7 then reduces to Idf(f)
0:
k't-h
(10)
in which k' = k,,,p". This expression allows the time dependence of the annihilation rate constant to be obtained without knowledge of the effective reaction order. The assumption that p ( t ) is constant may be relaxed by substituting I p ( t ) ,the phosphorescence intensity, for p ( t ) if a single triplet population is involved. When emission from both trapped and mobile excitations is included in Ip,this substitution will only be valid at early times, Le., when k a n n Pis much greater than k , . Within the appropriate time range, the time dependence of k can be obtained from
With the correct value of m, a plot of In (Idl/Ipm)vs In t is expected to be linear with slope -h. The molecularity of the reaction can therefore be obtained by substituting 1 and 2 for m and checking the linearity of the resulting In-In plots. The advantage of this approach is that it permits explicit account to be taken of the effects of the triplet lifetime. In a homogeneous medium, the assumption of (5) holds, with I p ( t ) substituted for p ( t ) . However, in heterogeneous materials, regions of material may be effectively segregated from one another, leading to isolated triplet populations. In such cases, the phosphorescence intensity does not represent only the reacting population and the analysis of Idf will be limited to time scales on which (IO) holds. it has been shown that, for k,,, = 0.1k , and h = 0.6, times shorter than 1O-)T are required to obtain h correctly within 2%. Increasing k,,, (relative to k , ) or h has the effect of further reducing the range of applicability of ( 1 I ) . 4.2. Exact Solutions. Four classes of reaction are described by (9), distinguished by the possible values of h and m: (i) classical heterofusion ( h = 0, m = 1); (ii) classical homofusion ( h = 0, m = 2); (iii) dispersive heterofusion ( h # 0, m = I ) ; and (iv) dispersive homofusion ( h # 0, m = 2 ) . Exact expressions of p ( t ) are obtained by solving (9) in each case. Given analytical expressions for p ( t ) , the corresponding expressions for I d f ( t ) can be obtained directly from (7). Such expressions permit analysis of annihilation reaction kinetics from delayed fluorescence decays alone, over the entire time range of observation. a. Classical Expressions for p ( t ) . In the simplest case, classical heterofusion, eq 9 reduces to dP
-- = ( k ' + kl)p
dt
with solutions ~ ( t =) PO expl-(k'ann
+ ki)tJ
(13)
in which k',", is the product of k,,, and p'. At all times, this is
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990 3451
Transport Kinetics in Vapor-Deposited Naphthalene TABLE I: Summarv of Expressions for d t ) and I d t )
nonclassical heterofusion ( h # 0, m = 1) po
exp(
I
-%
- klr)
kannpOrh exp --
- klt
:.":
I
+
a simple exponential with an effective rate constant k = k',,, ki. In case ii, classical homofusion, eq 9 may be rearranged to yield
-
dP = -dt P(kannp + k ~ )
(14)
which, upon integration, yields
At times short relative to the natural lifetime, T , the exponential in (1 5) may be expanded, and if (kannpo+ k l ) t >> 1, this yields p(t)
(kannPO + k l ) - l r - l
(16)
which formally resembles the solution to (14) in the absence of the natural decay term. b. Dispersive Expressions for p ( t ) . The general rate equation (9) is now examined with h # 0. For unary processes ( m = I ) , a separable equation is obtained dP = (kannt-' dt
--
+ kl)p B
with p ( t ) given by p ( r ) = po
exp
1
( 'rt: ) --
- kit
(18)
3
For the dispersive case in which annihilation is a binary process, (9) becomes 01
*
0.0
Unlike ( l 7 ) , this is not separable. However, after substitution of z = l / p , an equation in the following form is obtained: dz/dt
- PZ = Q, P
= k l , Q = kannrh
(20)
The general solution is given by23
Evaluating from time = 0 to t and substituting
p =
l / z yields
TABLE II: Rate Constants (in s) and Lifetimes (in s-I) of Delayed Fluorescence and Phosphorescence Decays Obtained from Data between 0.5 and 1.0 s 1
2 3
(23) Boyce, W. E.; DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems; Wiley: New York, 1969. (24) Abramovitz, M.; Stegun, I . A. Handbook of Mathematical Functions; Dover Publications: New York, 1970.
1.0
TME (SEC) Figure 2. Phosphorescence and delayed fluorescence decays of vapordeposited 100%naphthalene-h8. (a) In phosphorescence as a function of time. (b) In delayed fluorescence as a function of time. Straight lines represent least-squares fits of data after 0.5 s with slopes: (a) k , = 0.58 s-', 7, = 1.73 S; (b) kdf = 2.05 S-', 7 = 0.49 S.
sample
in which y is the incomplete gamma function.24 The relationship expressed in ( 7 ) between Idfjt) and p ( t ) allows one to readily obtain expressions for I d f j t ) from the corresponding expressions for p ( t ) . These results are summarized in Table I .
0.5
'
~ D F
2.05 2.18 2.01 1.42 (420 nm)
k, 0.58 0.61 0.63 (SWP) 0.57 (540 nm)
k ~ ~ l k TDF p 3.55 3.56 3.19 3.53
0.49 0.46 0.50
7P
1.73 1.63 1.58 (SWP) 1.75 (540 nm)
It should be noted that in all four cases both p and I d f exhibit simple exponential behavior in the long time Iimit.l3 Calculation of I d f ( t ) is trivial using the expressions of Table I except in case iv; in this particular case, the integral ~ ( f k ~has t )no analytical solution and must be evaluated by numerical integration. Alternatively, the gamma probability distribution, P ( a , x ) y ( a , x ) / I ' ( a ) ,and r ( a ) can be computed; MDGAM from the NAASIMSL library and GMMMA from NAAS:SSP, the IBM
Harmon and Kopelman
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
TABLE 111: Comparison of Different Models in Fitting Delayed Fluorescence: Outout Parameters and y 2 Values h m x2
a
Classical 1 2
2.38 0.00054
I 2
1.13 0.65
0 0
4.82 1.09
Diffusive, Unconstrained 0.49 0.49
0.005 1 0.005 1
0
* l -I0 $4.
-2 0
0
-r! 0
>
4 0
-I!0
LN (TIME/SEC)
i
b
N .
-8.0
c4. 0
Ln T I E
-2 0
0
'1 8; t4.
0
-a. o
b
-2.0
I
-a. 0
o +& 0
4b .0
u( T I Y
Figure 4. Intensity ratios as functions of time. (a) In (Idl/I,,) vs In time from 20 to 135 ms. (b) In &/I:) vs In time from 20 to 135 ms. Straight lines represent best fits of ratio I-*, with slopes -h. Values of h are 0:
(a) 0.49 and (b) 0.40.
isolated, obey the same power law as the delayed fluorescence. If an exponential component calculated by using the rate constants of Table 111 is subtracted from the phosphorescence, the remaining intensity also obeys this law. Since the natural lifetime of the *~ emission in this time regime excimer is about 100 n ~ , excimer must be a result of annihilation and is therefore expected to behave like the delayed fluorescence. The excimer component of the nominal phosphorescence decay complicates analysis of the kinetics using eq 1 1. In Figure 4,the intensity ratios of (1 1 ) are shown out to 135 ms. It appears that the result obtained with m = 2 is linear over a longer time range than with m = 1. This suggests that, regardless of the mechanism at the earliest times, annihilation is primarily a binary process after 100 ms. This observation is consistent with the analysis of the long time behavior (discussed below). However, the difference in fitting to unary and binary expressions is not pronounced enough to distinguish between the two mechanisms. This is attributed to contamination of phosphorescence by delayed excimer emission; phosphorescence intensities were obtained from a wide spectral region and do not accurately reflect the decay of p ( t ) . Classical kinetics predicts that the delayed fluorescence should decay as a power of the phosphorescence, reflecting the molecularity of the annihilation process. tdf
0:
I,X
x = 1, 2
( 2 5 ) Arden, W.; Peter, L. M.;Vaubel, G.J . Lumin. 1974, 9, 257.
(23)
The Journal of Physical Chemistry, Vol. 94, No. 9, I990 3459
Transport Kinetics in Vapor-Deposited Naphthalene a
a
II
0
b
N
N 0
Gk.00
Figure 5. Classical fit of experimental delayed fluorescence decay of vapor-deposited 100% naphthalene-hs: (a) experimental decay (small circles) with best fits to eq 13 (squares) and eq 15 (diamonds); (b) residuals of the fits in (a). A dashed line is drawn at zero for reference.
An analogous expression obtained from the dispersive binary rate law yields26
x=-2 - h
0120
1.0
TABLE IV: Parameters Obtained from Simplex Fits of Delayed Fluorescence Decavs m ki ( T ) h X2 k , (7) h
Sample 1
If the phosphorescence is exponential, I , = Ape-kp‘,the delayed fluorescence is also expected to be exponential with a rate constant
(25)
Values of the ratio k d f / k pare found to be about 3.5 (Table II), deviating significantly from the classical values of 1 or 2. By use of (24), h can be extracted from X:
h = -2 - x 1
0;so
Figure 6. Data of Figure 5 fit to eq 18 (squares) and eq 22 (diamonds). Symbols have been omitted from the fitted curve in (a) to leave the experimental points clear. Residuals are plotted in (b) on the same scale as Figure 5b to emphasize the difference.
I-h
kdf = X k ,
0.40
-x
The value of h corresponding to X = 3.5 is 0.60, which agrees with values calculated directly from the delayed fluorescence. Since ( 2 4 ) assumes a binary process, this agreement is further evidence that triplet-triplet annihilation at long times is best described as homofusion rather than heterofusion. 5.2. Fitting with Exact Solutions. Analysis of experimental decays over the full time range monitored requires the exact expressions derived in section 4.2. Delayed fluorescence decays were tested against the expressions in Table I with a simplex fitting procedure, as summarized in the Appendix. By this means, the factors h and k , were calculated simultaneously from the entire time range of the measured decay. In Table I, po appears only as a prefactor or multiplied by k,,, or kin,. po cannot be retrieved from the experimental data or justifiably treated as a parameter independent of k,,,. Therefore, the prefactors of po were incorporated into normalization factors and po was arbitrarily set to 1 during fitting. Typical results of fitting a single delayed fluorescence decay to the expressions derived here are presented in Table 111. The parameters with corresponding x 2 values resulting from fits to classical and dispersive uni- and bimolecular expressions are shown. For unary and binary cases, x2 is improved by a factor of 1000 and 200, respectively, when the dispersive rather than classical expression is used. This comparison is illustrated graphically in Figures 5 and 6. In Figure 5a, an experimental delayed fluorescence decay is plotted, together with curves corresponding (26) Anacker, L. W.; Kopelman, R. J . Chem. Phys. 1984, 81, 6402.
1 2
1.13 (0.88) 0.65 (1.5)
0.49 0.49
1 2
0.35 (2.9) 0.54 (1.9)
0.41 0.41
1 2
1.26 (0.79)
0.70 0.70
0.0051 0.0050
0.58 (1.7)
0.60
0.61 (1.8)
0.63
0.57 (1.8)
0.75
Sample 2 0.038 0.038
Sample 3 0.63 (1.6)
0.0030 0.0030
to the classical parameter sets of Table Ill; the residuals are plotted in Figure 5b. It is clear that neither of the classical expressions can reproduce the experimental decay. Fits of the same decay with the dispersive expressions are shown in Figure 6a, with the residuals below. The two fits are indistinguishable, both from the experimental decay and from each other. In Table 1V are presented the results of fitting six delayed fluorescence decays to the dispersive expressions in Table I with the simplex routine. The parameters h and k l were varied independently, and a relative error limit of was specified for search termination. In each case, all data between 20 and 1000 ms were fit, a total of 98 1 experimental points in each decay. The resulting x2 values are listed in the middle column. To the right are tabulated corresponding values obtained above (Table I and section 5.1). The value of h obtained from the fits is independent of the assumed molecularity, m. In addition, the fits, as measured by x2,are equally good for m = 1 and m = 2. There are striking differences, however, in the natural decay rates, predicted by the two mechanisms. In two of three cases, the value of k , necessary to match the experimental time dependence with a unary expression is incompatible with results obtained from the phosphorescence decays at long times. These fits must be rejected therefore as unphysical descriptions of the processes occurring in these samples. However, the fact that the same value of h is obtained for unary and binary models is indicative of a system in which both types of annihilation take place, so that neither molecularity alone is sufficient to describe the reaction kinetics. If the decays were carried out to longer times, x 2 would be more sensitive to k , and the relative merits of the uni- and bimolecular expressions might be more definitively as-
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The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
sessed. However, it is clear that eq 9, incorporating both natural decay and a time-dependent annihilation rate term, adequately describes the time evolution of the delayed fluorescence in these disordered naphthalene solids. Inspection of the sequence of parameters and x2 values generated during the search procedure suggested that x2 is a steep function of h and less strongly dependent on k , . T o further investigate this possibility, decays were fit by varying only k,,, and h, keeping k , constant. Values of k , were chosen between the effective k , (from the long-time phosphorescence tail) and the isolated molecule value. The effect of the natural lifetime is more pronounced in the bimolecular (m = 2) case: an increase of 50% in k , (0.4-0.6 s-') produced a 15% increase in h. For fits with m = 1 , the same increase in k , resulted in only a 2 4 % increase in h. These results show that the natural lifetime is less critical than h in determining the time dependence of the delayed fluorescence in this time regime ( I s), which is not surprising, considering that this corresponds to about 0.55. 6. Discussion
The kinetics of triplet excitation transport in disordered naphthalene has been examined through the triplet-triplet annihilation reaction rate coefficient, k ( t ) . Several methods of obtaining k ( t ) from experimental delayed fluorescence decays have been presented. The time dependence of k has been shown to have the power law dependence predicted by (6). However, the exponent of time, h. varies with the method used to derive it. Values of h are consistently higher when derived from the approximate equation ( I O ) rather than solutions to the full rate equation. This is explained by the fact that (10) assumes a constant triplet excitation density. Higher h values correspond to more rapid decays, reflecting the effects of annihilation and natural decay on the triplet population, even at early times. The use of phosphorescence decays to correct for the time dependence of p , as In the samples in ( 1 1 ), has proven successful in other studied here, however, the existence of large-scale domain heterogeneity and interfering emission from the naphthalene excimer minimizes the value of eq 1 1 in kinetic analyses. Experimental delayed fluorescence decays are best described by expressions derived from full analytic solutions of the rate law, eq 9. Only solutions with a time-dependent annihilation rate coefficient fit the experimental data, confirming that the transport is indeed dispersive.2,22 The values of the time exponent, h , are f m n d to be independent of the effective reaction molecularity assumed and vary from 0.4 to 0.7. The annihilation reaction appears to be a binary process. Good agreement is found between h values derived from long and short time behaviors (eq 25 and 10, respectively). The long time expression assumes a binary reaction, so, although no such assumption is made in (lo), a binary reaction is indicated. The unimolecular decay constant, k,, obtained from fits to exact solutions of eq 9, are more reasonable when a binary reaction is assumed. I t is probable, however, that both mechanisms operate to some extent, at least at early times, given the heterogeneity of these materials. As mentioned above, the same value of h characterizes the annihilation rate at early and long times. If this exponent is determined by an effective geometry of available sites, one would conclude that the system has reached a quasi-equilibrium distribution of occupied site energies before 20 ms, the earliest time of observation in these studies. Excitons persisting at longer times are partially immobilized in local traps, undergoing monomolecular decay but subject to thermal detrapping. This detrapping acts as a continuing source of excitations which can undergo annihilation. In this regime, monomolecular decay dominates the change in triplet density, p , but the power-law dependence of the annihilation rate, -(dp/dt),,, on p still holds. The underlying medium sampled by the excitons is the same in both regimes; this (27) Kopelman, R.; Li. C. S.; Parus, S. J.; Prasad, J. In Proceedings of Third International Conference on Unconventional Photoactive Solids, Schloss Elmau. W . Germanv, October 1987.
Harmon and Kopelman may explain why the exponent, h, remains unchanged. The values of h obtained suggest that excitation motion in this time regime is confined to quasi-one-dimensional regions, indicating a high degree of energetic disorder. This confirms that the reaction being observed is in fact occurring in or on the boundaries of type I1 regions of these materials. Random walk simulations on energy-disordered lattices have shown that relaxation occurs very rapidly, within the first 25-50 steps.28 This. too, is consistent with observation of the lowest energy sites of type 11 regions in the time regime reported here. Values of h have been obtained from delayed fluorescence and phosphorescence spectra of isotopic mixed ~ r y s t a l s . ~Below ~~.~~~~~ percolation, at dopant concentrations of 2-6%, h is found to be about 0.45 and relatively insensitive to concentration. This value is comparable to the values reported here, and we conclude that the portion of the samples in which the reaction is taking place is extremely ramified.30 Without postulating total correspondence between the two systems, it is reasonable to think that the set of sites comprising the tail of the energy distribution in structurally disordered samples is about as ramified as the set of dilute guest sites in low-concentration mixed crystals. However, the topology is also consistent with one-dimensional filaments,' for which h = 0.5. We note that the triplet energy transport topology in naphthalene crystals is effectively two-dimensional (in the ab plane).3.4.7,".'2 Thus, two-dimensional geometrical domain boundaries may give rise to effectively one-dimensional energy transport regions.15 From the results reported here, we also conclude that both natural decay and annihilation should be taken into account in the analysis of triplet-triplet annihilation kinetics whenever possible. I f any uncertainty exists as to the origin of the observed phosphorescence. the exact expressions of Table I11 should be employed. Acknowledgment. This work was supported by NSF Grant DMR 88-01 120.
Appendix: Fitting Procedure A simplex algorithm for nonlinear least-squares fitting was used to find the set of parameters, [k,,,,k,,h], which most closely reproduced the experimental delayed fluorescence decays. If one uses the notation Idf(tj) for the theoretical intensity at time t, calculated from the expressions in Table X and F(ti) for the experimental intensity at time ti, the x2 (chi-square) error of a given fit, as defined by D e m a ~ is, ~ ~ X*
i CWi[ldt(ti) i
- F(ti)12
(AI)
wi is a weighting factor which is set to 1 for the purposes of this work. Both Id[ and F a r e normalized over the time interval being fit, and the set of parameters that minimizes x2 defines the best fit. In brief, this is found by exploration of the surface defined by X2(k,,,,kl,h,ti) with a moving set of points in parameter space. Each point is generated from a single initial point by changing the value of one parameter. The n 1 points (in an n-parameter fit) are termed the simplex. x2 is calculated at each point, Le., for each set of parameters which in turn defines a point. By successively replacing the point with the largest x2 value, the parameter space is explored until a minimum in x2 is found. The criterion for search termination is that the difference in minimum x2values of the last two simplexes generated fall below a specified relative error limit (RERR), i.e.
+
(28) Anacker, L. W.; Harmon, L. A,; Kopelman, R. Unpublished results. (29) Anacker, L. W.; Klymko, P. W.; Kopelman. R. J . Lumin. 1984, 31/32, 648. (30) Kopelman, R. J . Stat. Phys. 1986, 42, 185. (3 I ) Demas, J. M. Excited Slate Lgetime Measurements; Academic Press: N e w York, 1983
J . Phys. Chem. 1990, 94, 3461-3466 The specific algorithm used follows that of D e ~ n a s , ~and ' the procedure is described fully in section D.l of that reference. The search method expands in regions of steeper x2 dependence and contracts as x 2 varies less rapidly. By this means, the surface in the vicinity of a minimum is searched most thoroughly. However, in contrast to grid methods, which locate minima with respect to a single parameter at a time, the entire parameter space need not necessarily be explored. Experimental decays were fit by first specifying a relative error limit of The parameters found by this search were used as input to a search with a relative error limit of IO4. In some cases, the parameters obtained with RERR = did not differ sig-
3461
nificantly from those found with RERR = 10-j; in others, a set of quite different parameters with a lower x2value was obtained with RERR = 1 O-4. Because some regions of parameter space may never be searched with the simplex method, it might be vulnerable to systematic biasing by initial guesses. To ensure that bias is avoided, several quite different initial parameter sets can be used, together with low relative error limits, in order to obtain parameters that can be used with confidence. In the case of the fits reported here, output parameters were found to be entirely independent of initial parameter^.'^ Registry No. Naphthalene, 91-20-3.
Pressure- and Temperature-Dependent 'H NMR Studies of N-Methylmorpholine Ring Inversion Clifford B. LeMaster, Carole L. LeMaster, Mohsen Tafazzoli, Cristina Suarez, and Nancy S. True* Department of Chemistry, University of California, Davis, California 95616 (Received: May 30, 1989; In Final Form: November 27, 1989)
Gas-phase 'HNMR spectra of N-methylmorpholine (MM) display exchange-broadened line shapes that are both temperature and pressure dependent. Infinite pressure, unimolecular gas-phase inversion rate constants are lower and the free energy of activation is higher than observed in liquid-phase samples. This phase dependence is consistent with an activation volume, A v',of -9 cm3 mol-' and a transition state smaller than the equilibrium chair conformation. Gas-phase unimolecular activation parameters for the chair to twist-boat inversion are E,,, = 52.5 (0.4) kJ mol-', A = 1.8 (0.3) X I O i 3 s-I, AC' = 49.9 (0.4) kJ mol-', AH' = 50.3 (0.4) kJ mol-', and AS' = 1.3 (2.6) J mol-] K-I. Activation parameters for a liquid sample containing 1.00 mol % MM in CS2 are AG' = 46.3 (0.4) kJ mol-', AH' = 47.3 (0.8) kJ mol-I, and AS' = 3.3 (3.3) J mol-' K-I. The pressure dependence of the inversion rate constants agrees with RRKM calculations, indicating stochastic intramolecular vibrational redistribution (IVR) in MM at internal energies of ca. 50.2 kJ mol-' and 9800 vibrational states/cm-I.
Introduction Our previous studies of six-membered r i n g ~ l -have ~ demonstrated that gas-phase NMR experiments can provide insight into the vibrational dynamics of conformationally converting molecules and, when combined with liquid studies, information about the nature and direction of solvent effects. We continue these studies with N-methylmorpholine, the first molecule we have investigated that contains both ring heteroatoms, oxygen, and nitrogen. As with the previous rings studied, rate constants were found to be pressure dependent. Comparison of the falloff of these rate constants with pressure with theoretical calculations based on RRKM kinetic theory supports statistical vibrational redistribution in MM at internal energies of ca. 50.2 kJ mol-' and state densities of 9800 states/cm-I. We have made a significant change in our method of performing RRKM rate calculations for this study. We have included the bath gas collisional efficiency as a separate adjustable parameter which is treated as a constant. Previously, this factor was included in the collision diameter, generally causing it to appear low versus hard-sphere estimates. Inclusion of the efficiency in the calculation affects the collisional frequency, w 8kbT ' I 2
-=[$][TI au2 ( I ) Ross, B. D.; True, N . S. J . Am. Chem. SOC.1983, 105, 4871-4875. (2) Chu, P. S.; True, N. S. J . Phys. Chem. 1985, 89, 5613-5616. (3) Chu, P. S.; True. N. S . J . Phys. Chem. 1985, 89, 2625-2630. (4) LeMaster. C. B.; LeMaster, C. L.; Tafazzoli, M.; Suarez, C.; True, N. S . J . Phys. Chem. 1988, 92, 5933-5936. ( 5 ) LeMaster, C. B.; LeMaster, C. L.; Suarez, C.; Tafazzoli, M.; True, N. S. J . Phys. Chem. 1989, 93, 3993-3996.
0022-3654/90/2094-346l$02.50/0
where kb is the Boltzmann constant, P the pressure, T the absolute temperature, p the reduced mass, and u the collision diameter. The three terms represent, from left to right, the concentration in molecules the molecular speed in cm s-I, and the cross section in cm2. The collisional efficiency can be used to adjust w to an effective collisional efficiency, weff, by use of eq 2, where
the subscripts represent quantities associated with collisions between two reactant molecules ( u l l , l l 'or ) a reactant and a bath gas molecule (uI2,p.,?),where u l I and u12are effective collision diameters6 The collisional energy-transfer effciency on a pressure per pressure basis, &,, defined in eq 3, may be determined ex-
perimentally by methods described previously.6-' These methods require determination of bimolecular rate constants at several pressures i n the bimolecular region for the pure molecule. Equation 3 illustrates that the experimentally determined quantity, Pp, inherently contains efficiency differences attributable to differences in the reduced masses and collision diameters of the bath gas versus the reacting molecule; therefore, the calculations can (6) Chan, S. C.; Rabinovitch, B. S.; Bryant, J. T.; Spicer, L. D.; Fujimoto, Y . ;Lin, N.; and Pavlou, S. P. J . Phys. Chem. 1970, 74, 316C-3176. The collisional energy transfer efficiency &, is reflected by the effective collision diameters u l , and uI2. (7) Chauvel, J . P., Jr.; Friedman, B. Ri.; Van, H.; Winegar, E. D.; True, N. S. J . Chem. Phys. 1985, 82, 3996-4006.
0 1990 American Chemical Society
